Cosmological sound speed ($c_s^2$)

For a barotropic fluid, the squared sound speed is $c_s^2=\partial P/\partial \rho$. A Fourier density mode obeysin linear Newtonian perturbation theory.

For a barotropic perfect fluid with constant $w$, its cosmological adiabatic sound speed satisfies $c_s^2=w$. If $w<0$, the short-wavelength pressure term has the wrong sign for restoring oscillations: a mode grows as $\exp (\sqrt{-w}k\tau )$ when expansion can be neglected. For $w=-1/3$ in a coasting background with constant conformal expansion rate $K$, the exponents are $-K\pm \sqrt{K^2+k^2/3}$. This is a gradient instability, not ordinary positive-pressure Jeans instability. A network with tension or elastic anisotropic stress needs different constitutive perturbation equations; a negative background equation of state alone does not establish its perturbation sound speed.

The squared adiabatic sound speed is the ratio of background pressure and energy-density time derivatives. For a barotropic equation of state it is $dP/d\rho$. The non-adiabatic pressure perturbation is the difference $\delta P-c_a^2\delta \rho$; in a general field or multicomponent model this background quantity need not equal the rest-frame propagation speed.